where \(D_{pp}\)  is the diffusion term in the momentum space, associated to Fermi-II acceleration processes, which we model as \(D_{pp}=\chi p^2 /4\)  so that the characteristic acceleration time is \(\tau _{acc}=p^2/4D_{pp}=\chi ^{-1}\)  [1]}. The energy loss term \(b_i(p)\)  is given by the sum of the Coulomb losses, \(b_{Coul}\propto n_{th}\)  [2]}, where \(n_{th}\)  is the density of the thermal gas, of the radiative losses (synchrotron and inverse Compton scattering), \(b_{rad}\propto (B^2+B^2_{ICS}) p^2\)  [3]}, and of the adiabatic expansion losses, \(b_{ad} \propto (dV/dt) p\) , where we assume that the volume is changing with a Sedov-like expansion \(V(t)\propto (t/t_0)^{6/5}\)  [4]}.
